A quantitative analysis of Galilei's observations of Jupiter satellites from the Sidereus Nuncius
We present a new careful and comprehensive analysis the observations of the satellites of Jupiter from the Sidereus Nuncius that extends and complements previous similar studies. Each observation is compared to the predictions obtained using a modern sky simulator, verifying and trying to understand them individually. The work considers both the information that can be extracted from the sketches and the angular measurements reported by Galilei. Angular measurements allow assessing the absolute accuracy in relation to modern ephemerides. We evaluate the performances of the telescope in terms of separation power of close-by satellites and the inefficiency in the detection connected to the proximity to the disk. A sinusoidal fit of the data, allows measuring the relative major semi-axes of the satellites' orbits and their periods with a statistical precision of 2-4\% and 0.1-0.3\% respectively. The posterior fit error is used to estimate the measurements precision. We show that with this data one can infer in a convincing way the third law of Kepler for the Jupiter system. The 1:2:4 orbital resonance between the periods of Io and Europa/Ganymede can be determined with \% precision. In order to obtain these results it is important to separate the four datasets. This operation was an extremely difficult task for Galilei. Nevertheless we show how some indication on the periods emerge from the using the modern Lomb-Scargle technique on the full data-set. We briefly extend the use of the simulator to verify the accuracy in the seven observations of the Moon and the performance in reproducing the Pleiades, the Orion belt, the Orion head and the Beehive cluster. Finally we present images obtained with a replica of the telescope that highlights the challenges of these observations thus confirming the excellence underlying this amazing set of early scientific data.